The Number of Points on Two Line Segments

We say that a set is countably infinite if we can pair the elements with set of counting numbers 1, 2, 3, and so on. Believe it or not, the number of positive integers and the number of integers (both negative and positive including 0) have the same number of elements. It is because we can pair them in a one-to-one correspondence such as shown in the below.

As shown on the table, if we continue indefinitely, we know that we can pair each counting number with an integer in a one-to-one correspondence without missing any element. Using this concept, we show intuitively that the number of points on two line segments is equal even if they have different lengths. We can do this by showing that for each point on segment $\overline{AB}$, there is a corresponding point on segment $\overline{CD}$. » Read more

Real Numbers: A Summary

For the past two years, we have talked a lot about real numbers.  We have talked about integers and its operations (addition, subtraction, multiplication, and division), we have discussed about rational and irrational numbers, and we have talked about their properties, structure, and wonders. In this post, we are going to summarize what we have learned about them. Figure 1 - The Number Line

The set of real numbers is the collection of all rational and irrational numbers. By convention, real numbers are represented by a line infinitely long where the positive real numbers are situated at the right hand side of 0, while the negative are at the left hand side. It is also important to note that for each point on the number line, there exists a corresponding real number equivalent to it, and for each real number, there is a corresponding point on the line that represents it. » Read more